Divisibility Rules and Counterexamples: Exploring Number Theory Concepts

In summary, the conversation involves three problems related to divisibility and proofs. The first problem states that if q is divisible by the sum of r and s, then either q is divisible by r or q is divisible by s. The second problem involves proving that if d is greater than 0, then (fd+ed) is equal to d(f,e). The third problem states that if a is divisible by b, then a^m is divisible by b^m, where a, b, and m are positive integers. The conversation also includes some thoughts and potential ways to approach the problems. One suggestion is to use the definition of divisibility, while another suggests using a contradiction. The notation in the first problem is unclear, and
  • #1
1+1=1
93
0
hey i have more problems that can really exercise the mind! here are 3.

1. prove if q is divisible by (r +s) then either q is divisible by r or q is divisible by s.

2. if d>0, (fd+ed) = d(f,e). proof.

3. a divisible by b => a^m divisible by b^m a,b,m are in Z+.

i think i have some thoughts and i will share.

1. say that (b+c) = ma, where m is in Z+. after that, i am guessing...

2. could use contradiction here. say that d=0, then show the proof? anyone has any takes on this?

3. there exist m and n s.t. bn=ma. lost after here.

anyone w/ information/thoughts please share.
 
Physics news on Phys.org
  • #2
For number 3, just apply the definition of "divisibility"... There is a k such that a = bk, and thus a^m = (bk)^m = b^m * k^m, and so a^m / b^m = k^m \in Z, hence b^m | a^m.
 
  • #3
what does the notation in 1 mean, are you talking about ideals?

the second follows, i believe, if you show the RHS divides the LHS and the LHS divides the RHS
 
  • #4
What does (a+b) denote ?
 
  • #5
Number 1: By (r+s), do you mean the sum of r and s? If so, a quick counterexample: 25 is divisible by (2+3) but it's not divisible by 2 or by 3.
 
  • #6
AlMacD said:
Number 1: By (r+s), do you mean the sum of r and s? If so, a quick counterexample: 25 is divisible by (2+3) but it's not divisible by 2 or by 3.
Bah, I was just about to put the exact same counter example up :wink:
 

1. What are some effective brain exercises for improving cognitive function?

Some effective brain exercises for improving cognitive function include puzzles, memory games, learning a new language, and playing strategic board games like chess.

2. How often should I do brain exercises?

It is recommended to do brain exercises at least 3-4 times a week for about 30 minutes each session. Consistency is key for seeing improvements in cognitive function.

3. Can brain exercises help prevent cognitive decline?

Yes, research has shown that regularly engaging in brain exercises can help prevent cognitive decline and decrease the risk of developing age-related cognitive diseases such as Alzheimer's.

4. Are there any brain exercises specifically for improving memory?

Yes, some brain exercises that can help improve memory include association games, recalling events in detail, and creating mental images of information to be remembered.

5. Are there any negative side effects to doing brain exercises?

No, there are no known negative side effects to doing brain exercises. In fact, they have been shown to have numerous benefits for overall brain health and function.

Similar threads

  • Linear and Abstract Algebra
Replies
11
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
862
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
11
Views
2K
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
891
  • Linear and Abstract Algebra
Replies
1
Views
708
  • Introductory Physics Homework Help
Replies
6
Views
606
Replies
2
Views
634
  • Calculus and Beyond Homework Help
Replies
9
Views
699
Back
Top